
SIGMA 10 (2014), 112, 6 pages arXiv:1407.8291
https://doi.org/10.3842/SIGMA.2014.112
Configurations of Points and the Symplectic BerryRobbins Problem
Joseph Malkoun
Department of Mathematics and Statistics, Notre Dame UniversityLouaize, Lebanon
Received August 23, 2014, in final form December 17, 2014; Published online December 19, 2014
Abstract
We present a new problem on configurations of points, which is a new version of a similar problem by Atiyah and Sutcliffe, except it is related to the Lie group $\operatorname{Sp}(n)$, instead of the Lie group $\operatorname{U}(n)$. Denote by $\mathfrak{h}$ a Cartan algebra of $\operatorname{Sp}(n)$, and $\Delta$ the union of the zero sets of the roots of $\operatorname{Sp}(n)$ tensored with $\mathbb{R}^3$, each being a map from $\mathfrak{h} \otimes \mathbb{R}^3 \to \mathbb{R}^3$. We wish to construct a map $(\mathfrak{h} \otimes \mathbb{R}^3) \backslash \Delta \to \operatorname{Sp}(n)/T^n$ which is equivariant under the action of the Weyl group $W_n$ of $\operatorname{Sp}(n)$ (the symplectic BerryRobbins problem). Here, the target space is the flag manifold of $\operatorname{Sp}(n)$, and $T^n$ is the diagonal $n$torus. The existence of such a map was proved by Atiyah and Bielawski in a more general context. We present an explicit smooth candidate for such an equivariant map, which would be a genuine map provided a certain linear independence conjecture holds. We prove the linear independence conjecture for $n=2$.
Key words:
configurations of points; symplectic; BerryRobbins problem; equivariant map; AtiyahSutcliffe problem.
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References

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